Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
| direct product | #127×D21 |
| Fitting | C3×C7×#127 ⋊ C2 |
| Hall | C3×C7 ⋊ C2×#127 |
| Sylow | C7 ⋊ D3×#127 |
| Sylow | C3 ⋊ D7×#127 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C7×#127 | C2 |
| hypercenter | #127 | D21 |
| hypocenter | C3×C7 | C2×#127 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ #127 | |
996 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×#127, #127×C3⋊AGL(1,𝔽7), D21×#453, … |
| maximal normal subgroup of | D3×D7×#127, #127×C3⋊AGL(1,𝔽7), D21×#453, … |
| maximal quotient of | #127×C3⋊D21 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11094);